Mr Hancock, who is married, resigned the day after photos and videos of him kissing his part-time adviser, Gina Coladangelo, in his office were published by The Sun newspaper.
But Mr Murray said the UK Government still faced “major questions” about conflicts of interest, government contracts and the use of personal email accounts despite Mr Hancock’s resignation.
Speaking on the BBC Good Morning Scotland programme on Monday, the Edinburgh South MP said: “It’s not just breaking the Covid rules, it’s not just the affair, it’s who planted secret cameras in the Secretary of State’s office, why has he being using private emails, who sold it to the newspapers?
“All of that then leads into the big issues around the cronyism of PPE contracts. You were more likely to get a contract for PPE in the NHS if you knew Matthew Hancock rather whether or not you could actually supply and were an expert in it.”
Calling for an independent inquiry into the saga, Mr Murray said: “There are huge problems in this government in terms of its secrecy and in terms of the way operates – it doesn’t abide by any of the rules that normal Governments would abide by and that has to be investigated at the very highest levels.”
He added: “The affair is just the tip of the iceberg.
“In terms of breaking the Covid rules with social distance, it is only a part of the story here.
“The huge part of the story is all the issues that remain unresolved with regards to cronyism and contracts for many hundreds of millions of pounds going to companies associated with Matthew Hancock, whether it was an ex-neighbour or a friend.”
The issue of ministers using private emails for official business – in breach of government rules – resurfaced after revelations by The Sunday Times.
According to The Sunday Times, leaked documents show the former health secretary may have hidden details of their official dealings and his wider conduct in office.
In the minutes of a meeting between senior officials at the Department of Health and Social Care (DHSC) last December, the department’s second permanent secretary, David Williams, said Mr Hancock “only” deals with his private office “via Gmail account”.
However, the Department for Health has claimed all ministers “only conduct government business through their departmental email addresses”.
Questions have also been raised about Mr Hancock’s involvement in helping his friend, and former landlord of his local pub, obtain a multi-million pound contract to supply test tubes for NHS Covid-19 testing.
When Mr Hancock was asked about the deal at a No.10 coronavirus press conference in December, he replied: “I had absolutely nothing to do with that contract.”
But the Mail on Sunday, following a contested Freedom of Information request, obtained messages between Mr Hancock and Alex Bourne in which he personally referred a plea for business to Jonathan Marron, the-then director general of community and social care at the Department of Health.
A message from Mr Bourne initially raised the possibility of making personal protective equipment (PPE) such as surgical facemasks before later exchanges reveal a switch to producing items involved in coronavirus testing.